To deal with infinite *x*,
we will now extend the treatment of semimeasures on *B*^{*} in the
previous subsection by discussing
probability distributions on
.

**Proof.** The following proof is due to M. Hutter
(personal communications by email following a discussion of
enumerable and approximable universes on 2 August 2000 in
Munich). It is an extension of a modified^{1} proof [#!LiVitanyi:97!#, p. 249 ff]
that there is no universal recursive semimeasure.

It suffices to focus on .
Identify strings with
integers, and assume *P*(*x*) is a universal approximable
semidistribution. We construct an approximable semidistribution
*Q*(*x*) that is not dominated by *P*(*x*), thus contradicting the
assumption. Let
be a sequence of
recursive functions converging to *P*(*x*). We recursively define a
sequence
converging to *Q*(*x*). The basic
idea is: each contribution to *Q* is the sum of *n* consecutive
*P* probabilities (*n* increasing). Define *Q*_{0}(*x*):=0;
.
Let *n* be such that
.
Define *j*_{t}^{n} (*k*_{t}^{n}) as the element with
smallest *P*_{t} (largest *Q*_{t-1}) probability in this interval,
i.e.,
.
If
is less than twice and
is more than half of
*Q*_{t-1}(*k*_{t}^{n}), set
*Q*_{t}(*x*)=*Q*_{t-1}(*x*). Otherwise set
for *x*=*j*_{t}^{n} and *Q*_{t}(*x*)=0 for
.
*Q*_{t}(*x*) is
obviously total recursive and non-negative. Since
,
we have

Summing over

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